Cognitive Phase Transitions in Subjective Physics: Modeling Synchronization and Order Parameters with Reproducible Simulations
This repository is the official companion code for the paper:
Vladimir Khomyakov (2025). Cognitive Phase Transitions in Subjective Physics: Modeling Synchronization and Order Parameters with Reproducible Simulations. Zenodo. https://doi.org/10.5281/zenodo.17011035
It provides a complete implementation of the cognitive phase transition framework described in the paper, enabling reproducible simulations and analysis of synchronization dynamics in adaptive networks.
This work investigates cognitive phase transitions within the framework of Subjective Physics, providing direct computational evidence of a critical transition in an adaptive network of N=40 cognitive agents. We explicitly distinguish between the global order parameter |⟨ψ⟩| (Kuramoto mean-field amplitude) and the Mean Pairwise Coherence (MPC), a measure of local synchronization.
A clear non-monotonic transition was detected at control parameter r_c = 1.534, characterized by a sharp change (|ΔMPC| = 0.133) indicating a structural reorganization of the system's synchronized clusters. Following a transient period of desynchronization (MPC minimum ∼0.42), the system stabilizes into a high-coherence phase (post-transition MPC = 0.992 ± 0.007). The results demonstrate a phenomenology analogous to second-order phase transitions in physical systems, confirming key hypotheses of Subjective Physics regarding the reorganization of observer states.
cognitive_phase_transitions/
│
├── scripts/
│ └── cognitive_phase_transitions.py # Main simulation script
├── figures/ # Directory for saving output figures
│ ├── simulation_results.zip
│ ├── simulation_results.z01
│ ├── simulation_results.z02
│ ├── final_weight_matrix.pdf
│ ├── mpc_change.pdf
│ ├── network_phase_coloring.pdf
│ ├── order_parameter_and_control.pdf
│ ├── phase_transition_curve.pdf
│ └── synchronization_measure.pdf
├── requirements.txt # Python dependencies
├── .zenodo.json
├── CITATION.cff
├── LICENSE # MIT License
└── README.md # This file
Below is a complete description of all scripts included in this version. These files form a self-contained and reproducible package supporting the simulations, figures, and data analyses presented in the article.
cognitive_phase_transitions.py — Core simulation framework:
- Implements adaptive network of phase oscillators
- Combines Kuramoto-like synchronization with Hebbian plasticity
- Includes online covariance estimation for free-energy minimization
- Features transition detection and consolidation mechanisms
To run the cognitive phase transition simulations, you need Python ≥ 3.9. The required libraries can be installed via pip.
Create a virtual environment (recommended):
python -m venv cpt-env
# On macOS/Linux:
source cpt-env/bin/activate
# On Windows (PowerShell):
cpt-env\Scripts\activate
# On Windows (CMD):
cpt-env\Scripts\activate.batInstall dependencies:
pip install -r requirements.txtAlternatively, you can install them manually:
pip install numpy networkx matplotlib scipyRunning the simulation:
# Run the comprehensive simulation from the repository root directory
python scripts/cognitive_phase_transitions.pyThis will generate output showing:
- Phase transition detection at critical parameter values
- Evolution of order parameters and synchronization metrics
- Final network state visualization
- Statistical analysis of transition characteristics
Running the script cognitive_phase_transitions.py generates the following figures, which are central to the analysis in the accompanying article:
-
final_weight_matrix.pdf
Visualizes the final adaptive weight matrix ( W_{ij} ). The pattern reflects the history of the system's dynamics, with stronger connections consolidating the globally synchronized state achieved post-transition.
Corresponds to: Figure 8. Final Weight Matrix in the article. -
mpc_change.pdf
Plots the temporal change in Mean Pairwise Coherence (( \Delta\text{MPC} )). The phase transition is triggered when ( |\Delta\text{MPC}| > 0.1 ). The largest change (( |\Delta\text{MPC}|=0.133 )) occurs at ( t=413.70 ).
Corresponds to: Figure 4. Change in Synchronization in the article. -
network_phase_coloring.pdf
Displays the network structure with node coloring representing phase angles (final state, ( t=1000 )). Demonstrates the high degree of global synchronization achieved after the transition and subsequent stabilization.
Corresponds to: Figure 6. Network Structure and Final State in the article. -
order_parameter_and_control.pdf
Shows the dynamics of the global order parameter ( |\langle \psi \rangle| ) and the control parameter ( r ). ( |\langle \psi \rangle| ) remains negligible until the system exits the multi-cluster state after the transition, then increases steadily as global coherence is established.
Corresponds to: Figure 2. Dynamics of Global Order and Control in the article. -
phase_transition_curve.pdf
Illustrates the phase transition curve: Mean Pairwise Coherence (MPC) as a function of the control parameter ( r ) (smoothed with a moving average). Highlights the critical region near ( r \approx 1.5 ), the subsequent valley, and the final stabilization at high MPC for ( r > 2.0 ).
Corresponds to: Figure 3. Phase Transition Curve in the article. -
synchronization_measure.pdf
Charts the evolution of Mean Pairwise Coherence (MPC). The phase transition is detected at ( t=413.70 ). Shows the non-monotonic trajectory featuring a desynchronization valley before final stabilization.
Corresponds to: Figure 1. Evolution of Mean Pairwise Coherence in the article.
For complete scientific reproducibility, the raw numerical data used to generate all figures is preserved in a compressed split archive:
simulation_results.zip(17.4 MB) - Main archive partsimulation_results.z01(21.0 MB) - Archive part 1simulation_results.z02(21.0 MB) - Archive part 2
The archive contains the file simulation_results.npz (62.5 MB uncompressed) which includes:
psi_history- Complete history of complex field values ψ for all N=40 agents across all timestepsorder_param_history- Evolution of the global order parameter |⟨ψ⟩|r_history- Values of the control parameter r at each timestepW- Final weight matrix after simulation completion
To reconstruct the original simulation_results.npz file:
- Ensure all three archive parts are in the same directory
- Use any archive manager that supports split ZIP files (e.g., 7-Zip, WinRAR)
- Extract using the main file:
simulation_results.zip - The complete
simulation_results.npzwill be restored to its original 62.5 MB size
Note: This split archive format was necessary due to GitHub's 25 MB file size limit. The NPZ file contains the complete simulation data needed to verify all results and regenerate any figure independently.
# Run the comprehensive simulation
python scripts/cognitive_phase_transitions.pyThis will generate output showing:
- Phase transition detection at critical parameter values
- Evolution of order parameters and synchronization metrics
- Final network state visualization
- Statistical analysis of transition characteristics
Running the code will demonstrate:
Cognitive Phase Transition: The system undergoes a non-monotonic transition at r_c = 1.534, characterized by cluster reorganization.
Order Parameter Dynamics: Distinct behavior of global order parameter |⟨ψ⟩| and local synchronization measure (MPC).
Desynchronization Valley: Transient decrease in coherence (MPC minimum ∼0.42) during structural reconfiguration.
Stabilization: Final convergence to high-coherence phase (MPC = 0.992 ± 0.007).
Network Visualization: Final network state with phase coloring and weight matrix patterns.
If you use this model or code in your research, please cite the original publication:
@misc{khomyakov_vladimir_2025_17011035,
author = {Khomyakov, Vladimir},
title = {Cognitive Phase Transitions in Subjective Physics: Modeling Synchronization and Order Parameters with Reproducible Simulations},
month = aug,
year = 2025,
publisher = {Zenodo},
version = {1.0},
doi = {10.5281/zenodo.17011035},
url = {https://doi.org/10.5281/zenodo.17011035}
}- Version-specific DOI: 10.5281/zenodo.17011035
- Concept DOI (latest version): 10.5281/zenodo.17011034
- Download PDF: Direct link to paper on Zenodo
Theoretical Foundation: This work is based on the principles of Subjective Physics and builds upon the "Minimal Model of Cognitive Projection" (DOI: 10.5281/zenodo.16888675).
Methodological Inspiration: The framework combines elements from Ginzburg-Landau theory, Kuramoto synchronization models, and Hebbian/STDP-like synaptic plasticity.
subjective physics, cognitive phase transitions, synchronization, order parameters, kuramoto model, adaptive networks, reproducible simulations, observer entropy, collective intelligence, information theory
This project is licensed under the MIT License. See the LICENSE file for details.
This work builds upon the hypothesis of Subjective Physics formulated by Alexander Kaminsky. Special thanks to the researchers whose work on synchronization dynamics and phase transitions has inspired this computational framework.